All of the 5th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$36.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$15.00$ each for teachers and $$12.50$ each for students, and the group paid $$120.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 36}$ ${15x+12.5y = 120}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-15x-10.5y = -108}$ ${15x+12.5y = 120}$ Add the top and bottom equations together. $ 2y = 12 $ $ y = \dfrac{12}{2}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5x+3.5y = 36}$ to find $x$ ${5x + 3.5}{(6)}{= 36}$ $5x+21 = 36$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 6}$ into $ {15x+12.5y = 120}$ and get the same answer for $x$ ${15x + 12.5}{(6)}{= 120}$ ${x = 3}$ There were $3$ teachers and $6$ students on the field trips.